Pythagorean Theorem Calculator

Continue in the tool

More pages for this tool

find leg right triangle

Find a Leg of a Right Triangle

To find one leg of a right triangle from the hypotenuse c and the other leg b, isolate a = √(c² − b²). The tool requires c strictly greater than b: if c ≤ b, the value inside the square root would be zero or negative, with no real solution.

Isolating a leg in the Pythagorean equation

The formula a = √(c² − b²) comes from the same identity as the Pythagorean theorem, a² + b² = c², just isolating the unknown leg instead of the hypotenuse. That is why the sign inside the root is always a subtraction, unlike the hypotenuse calculation, which adds the two legs squared. For c=13 and b=12: a² = 169 − 144 = 25, so a = 5, forming the triple (5, 12, 13). For c=10 and b=6: a² = 100 − 36 = 64, so a = 8, forming the triple (6, 8, 10), exactly double the (3, 4, 5) triple.
The calculator rejects inputs where the hypotenuse is not greater than the given leg, returning an error instead of an invalid number: if c=5 and b=6 were entered, the calculation would need a² = 25 − 36 = −11, and the square root of a negative number does not exist among the reals. That c > b restriction is why, of the three Pythagorean theorem sides, only the hypotenuse can be the largest value: the two legs are always smaller than it, never equal to or greater.
Not every resulting leg is a whole number. For c=7 and b=5: a² = 49 − 25 = 24, and a = √24 ≈ 4.898979486, an irrational number the calculator displays with up to 10 significant digits. This happens whenever c² − b² is not a perfect square; only specific combinations, such as (5,12,13), (6,8,10) or (8,15,17), give three whole-number sides at once, called Pythagorean triples.

Examples

c=13, b=12

Input: c=13, b=12
Expected output: a=5

a² = 169−144 = 25; the (5,12,13) triple is the best-known primitive triple after (3,4,5).

c=10, b=6

Input: c=10, b=6
Expected output: a=8

a² = 100−36 = 64; the (6,8,10) triple is (3,4,5) multiplied by 2, so it stays Pythagorean.

c=7, b=5

Input: c=7, b=5
Expected output: a ≈ 4.898979486

a² = 49−25 = 24, which is not a perfect square; that is why this leg comes out irrational, unlike the previous examples.

Full tool FAQ

What is the Pythagorean theorem?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: c² = a² + b². Attributed to Greek mathematician Pythagoras (~570–495 BC).

What is the hypotenuse?

What are the legs?

What is a Pythagorean triple?

How can I check whether a triangle is a right triangle?

Does the tool work with decimal values?

How precise is the result?

Does the theorem apply to non-right triangles?

Frequently asked questions

Why does the hypotenuse need to be larger than the leg I already know?

Because the formula subtracts the known leg's square from the hypotenuse's square (a²=c²−b²); if c were smaller than or equal to b, the value inside the root would be zero or negative, so the calculator blocks that input instead of showing a numeric error.

Is the leg found always a whole number?

No. It is only a whole number when the three sides form a Pythagorean triple, like (5,12,13) or (6,8,10). In other cases, such as c=7 and b=5, the result is irrational (a ≈ 4.899) and shows as a decimal.

How do I know if the result formed a Pythagorean triple?

Check whether all three sides are positive whole numbers that satisfy a²+b²=c² exactly, without rounding; in the (5,12,13) and (6,8,10) examples they do, but for c=7, b=5 the leg of ≈4.899 is not a whole number, so it is not a triple.

Next steps

Continue in the tool

Open full tool

find leg right triangle

Find a Leg of a Right Triangle

Isolating a leg in the Pythagorean equation

The formula a = √(c² − b²) comes from the same identity as the Pythagorean theorem, a² + b² = c², just isolating the unknown leg instead of the hypotenuse. That is why the sign inside the root is always a subtraction, unlike the hypotenuse calculation, which adds the two legs squared. For c=13 and b=12: a² = 169 − 144 = 25, so a = 5, forming the triple (5, 12, 13). For c=10 and b=6: a² = 100 − 36 = 64, so a = 8, forming the triple (6, 8, 10), exactly double the (3, 4, 5) triple.

The calculator rejects inputs where the hypotenuse is not greater than the given leg, returning an error instead of an invalid number: if c=5 and b=6 were entered, the calculation would need a² = 25 − 36 = −11, and the square root of a negative number does not exist among the reals. That c > b restriction is why, of the three Pythagorean theorem sides, only the hypotenuse can be the largest value: the two legs are always smaller than it, never equal to or greater.

Not every resulting leg is a whole number. For c=7 and b=5: a² = 49 − 25 = 24, and a = √24 ≈ 4.898979486, an irrational number the calculator displays with up to 10 significant digits. This happens whenever c² − b² is not a perfect square; only specific combinations, such as (5,12,13), (6,8,10) or (8,15,17), give three whole-number sides at once, called Pythagorean triples.

Examples

c=13, b=12

Input: c=13, b=12
Expected output: a=5

a² = 169−144 = 25; the (5,12,13) triple is the best-known primitive triple after (3,4,5).

c=10, b=6

Input: c=10, b=6
Expected output: a=8

a² = 100−36 = 64; the (6,8,10) triple is (3,4,5) multiplied by 2, so it stays Pythagorean.

c=7, b=5

Input: c=7, b=5
Expected output: a ≈ 4.898979486

a² = 49−25 = 24, which is not a perfect square; that is why this leg comes out irrational, unlike the previous examples.

Full tool FAQ

What is the Pythagorean theorem?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: c² = a² + b². Attributed to Greek mathematician Pythagoras (~570–495 BC).

What is the hypotenuse?

What are the legs?

What is a Pythagorean triple?

How can I check whether a triangle is a right triangle?

Does the tool work with decimal values?

How precise is the result?

Does the theorem apply to non-right triangles?

Frequently asked questions